#124 - Ordered Radicals
The radical of \(n\), \(\operatorname{rad}(n)\), is the product of the distinct prime factors of \(n\). For example, \(504 = 2^3 \times 3^2 \times 7\), so \(\operatorname{rad}(504) = 2 \times 3 \times 7 = 42\).
If we calculate \(\operatorname{rad}(n)\) for \(1 \le n \le 10\), then sort them on \(\operatorname{rad}(n)\), and sorting on \(n\) if the radical values are equal, we get:
| Unsorted | Sorted | ||||
|---|---|---|---|---|---|
| n | rad(n) | n | rad(n) | k | |
| 1 | 1 | 1 | 1 | 1 | |
| 2 | 2 | 2 | 2 | 2 | |
| 3 | 3 | 4 | 2 | 3 | |
| 4 | 2 | 8 | 2 | 4 | |
| 5 | 5 | 3 | 3 | 5 | |
| 6 | 6 | 9 | 3 | 6 | |
| 7 | 7 | 5 | 5 | 7 | |
| 8 | 2 | 6 | 6 | 8 | |
| 9 | 3 | 7 | 7 | 9 | |
| 10 | 10 | 10 | 10 | 10 | |
Let \(E(k)\) be the \(k\)-th element in the sorted \(n\) column; for example, \(E(4) = 8\) and \(E(6) = 9\).
If \(\operatorname{rad}(n)\) is sorted for \(1 \le n \le 100000\), find \(E(10000)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=124. Published Friday, 14th July 2006, 06:00 pm. Solved by 15,439 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.2 Primes, Sieves, and Integer Factorization
Concepts: number-theory
Likely techniques: factorization hashing sorting
Learning mode
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Understand the problem
- What exactly is the input to problem 124? Is it a bound (10000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What do the arguments of E(k), E(4) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10000 and the cost of testing one.
- Which number-theory fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says number-theory / factorization - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10000, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example, 504 = 2^3 x 3^2 x 7, so rad(504) = 2 x 3 x 7 = 42.") - then run it. A surprise here is worth more than an hour of debugging later.
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- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
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- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
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Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the number-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the factorization idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 10000 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'factorization' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: explore the same concept filter in the Lab
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- Spaced re-attempt: come back after the review interval and re-solve it with no hints.